Nonimaging optical illumination system

ABSTRACT

A nonimaging illumination optical device for producing a selected far field illuminance over an angular range. The optical device includes a light source 102, a light reflecting surface 108, and a family of light edge rays defined along a reference line 104 with the reflecting surface 108 defined in terms of the reference line 104 as a parametric function R(t) where t is a scalar parameter position and R(t)=k(t)+Du(t) where k(t) is a parameterization of the reference line 104, and D is a distance from a point on the reference line 104 to the reflection surface 108 along the desired edge ray through the point.

Certain rights in this invention are retained by the U.S. Governmentpursuant to contract DE FG02-87ER 13726 of the U.S. Department ofEnergy.

This is a continuation-in-part of copending applications Ser. No.07/732,982, filed Jul. 19, 1991 and Ser. No. 07/774,666, filed Oct. 11,1991.

The present invention is directed generally to a method and apparatusfor providing user selected nonimaging optical outputs from varioustypes of electromagnetic energy sources. More particularly, theinvention is directed to a method and apparatus wherein the designprofile of an optical apparatus for extended optical sources can bedetermined by controlling the shape of the reflector surface to a familyof edge rays while simultaneously controlling the full contour of thereflected source. By permitting such a functional dependence, thenonimaging output can be well controlled using various different typesof light sources.

Methods and apparatus concerning illumination by light sources are setforth in a number of U.S. patents including, for example, U.S. Pat. No.3,957,031; 4,240,692; 4,359,265; 4,387,961; 4,483,007; 4,114,592;4,130,107; 4,237,332; 4,230,095; 3,923,381; 4,002,499; 4,045,246;4,912,614 and 4,003,638 all of which are incorporated by referenceherein. In one of these patents the nonimaging illumination performancewas enhanced by requiring the optical design to have the reflectorconstrained to begin on the emitting surface of the optical source.However, in practice such a design was impractical to implement due tothe very high temperatures developed by optical sources, such asinfrared lamps, and because of the thick protective layers or glassenvelopes required on the optical source. In other designs it isrequired that the optical source be separated substantial distances fromthe optical source. In addition, when the optical source is smallcompared to other parameters of the problem, the prior art methods whichuse the approach designed for finite size sources provide a nonimagingoutput which is not well controlled; and this results in less than idealillumination. Substantial difficulties arise when a particularillumination output is sought but cannot be achieved due to limitationsin optical design. These designs are currently constrained by theteachings of the prior art that one cannot utilize certain light sourcesto produce particular selectable illumination output over angle.

It is therefore an object of the invention to provide an improved methodand apparatus for producing a user selected nonimaging optical outputfrom any one of a number of different light sources.

It is another object of the invention to provide a novel method andapparatus for providing user selected nonimaging optical output of lightenergy from optical designs by controlling edge rays of a reflectedlight source.

It is a further object of the invention to provide an improved opticalapparatus and method of design wherein the reflector surface is tailoredto a family of edge rays.

It is a further object of the invention to provide an improved opticalapparatus and method of design for radiation collection.

It is yet another object of the invention to provide a novel opticaldevice and method for producing a user selected intensity output bysimultaneously controlling the full contour of a reflected source andtailoring the reflector to a family of edge rays.

It is still an additional object of the invention to provide an improvedmethod and apparatus for providing a nonimaging optical illuminationsystem which generates a substantially uniform optical output over awide range of output angles for finite size light sources.

Other objects, features and advantages of the present invention will beapparent from the following description of the preferred embodimentsthereof, taken in conjunction with the accompanying drawings describedbelow wherein like elements have like numerals throughout the severalviews.

DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a two-dimensional optical device for providing nonimagingoutput;

FIG. 2 illustrates a portion of the optical device of FIG. 1 associatedwith the optical source and immediate reflecting surface of the device.

FIG. 3A illustrates a bottom portion of an optical system and FIG. 3Bshows the involute portion of the reflecting surface with selectedcritical design dimensions and angular design parameters associated withthe source;

FIG. 4A shows a perspective view of a three-dimensional optical systemfor nonimaging illumination and FIG. 4B illustrates a portion of theoptical system of FIG. 4A;

FIG. 5 A shows intensity contours for an embodiment of the invention andFIG. 5B illustrates nonimaging intensity output contours from a priorart optical design;

FIG. 6A shows a schematic of a two dimensional Lambertian source givinga cos³ θ illuminance distribution; FIG. 6B shows a planar light sourcewith the Lambertian source of FIG. 6A; FIG. 6C illustrates the geometryof a nonimaging reflector providing uniform illuminance to θ=40° for thesource of FIG. 6A; and FIG. 6D illustrates a three dimensionalLambertian source giving a cos⁴ θ illuminance distribution;

FIG. 7A shows a two dimensional solution ray trace analysis and FIG. 7Billustrates three empirical fits to the three dimensional solution;

FIG. 8 shows an acceptance angle function which produces a constantirradiance on a distant plane from a narrow one-sided lambertian stripsource (2D) with a=1;

FIG. 9 illustrates a reflector profile which produces a constantirradiance on a distant plane from a one-sided lambertian strip source(2D) at the origin, R(φ=π/2)=1, a=1. CEC (inner curve) and CHC-typesolutions (outer truncated curve) are shown;

FIG. 10 shows a reflector designed to produce a reflected image adjacentto the source; the combined intensity radiated in the direction -θ isdetermined by the separation of the two edge rays: R sin 2α;

FIG. 11 illustrates an acceptance angle function which produces aconstant irradiance on a distant plane from a finite one-sidedlambertian strip source; there is only a CHC-type solution;

FIG. 12 shows a reflector profile which produces a constant irradianceon a distant plane from a finite one-side lambertian strip source ofwidth two units; note that there is only a CHC-type solution and it istruncated;

FIG. 13 illustrates a deviation of the reflector depicted in FIG. 12from a true V-trough;

FIG. 14 shows a desired irradiance distribution on a distant planeperpendicular to the optical plane divided by the irradiance producedalong the axis by the source alone; a broken line shows the irradianceof a truncated device;

FIG. 15 illustrates an angular power distribution corresponding to theirradiance distribution shown in FIG. 13; a broken line refers to atruncated device;

FIG. 16 shows an acceptance angle function corresponding to the desiredirradiance distribution plotted in FIG. 13;

FIG. 17 illustrates a reflector profile which produces the desiredirradiance shown in FIG. 13 on a distant plane from a finite one-sidedlambertian strip source of width two units; note that there is only aCHC-type solution and it is truncated;

FIG. 18 shows the slope of the reflector as a function of verticaldistance from the source;

FIG. 19 illustrates the deviation of the reflector depicted in FIG. 16from a true V-trough;

FIG. 20 shows the effect of truncation indicated by the angle up towhich the truncated device matches the desired power distribution, andplotted as a function of the vertical length of the reflector;

FIG. 21 illustrates a light source and family of edge rays given along areference line with identifying vectors;

FIG. 22A illustrates a source, reflector, reference line and edge raysfor a CEC reflector and FIG. 22B is for a CHC reflector;

FIG. 23 illustrates the effect of termination of the reflector onboundary illumination;

FIG. 24 shows a reflector for illumination of both sides of a targetzone;

FIG. 25 shows irradiance as a function of angle on a distant plane froma finite cylindrical source of uniform brightness;

FIG. 26 shows a CEC-type reflector profile producing a constantirradiance on a distant plane from a cylindrical source; and

FIG. 27 shows some edge rays corresponding to the angles designated inFIG. 25.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS A. Small Optical Sources

In the design of optical systems for providing nonimaging illuminationusing optical sources which are small relative to other systemparameters, one should consider the limiting case where the source hasno extent. That is, for example, the size of the source is much lessthan the closest distance of approach to any reflective or refractivecomponent. Thus the angle subtended by the source at any reflective orrefractive component may be regarded as small. Our approximation ofsmall source dimension d and large observer distance D amounts to d<<R₀<<D. This is in a sense the opposite of the usual nonimaging problemwhere the finite size and specific shape of the source is critical indetermining the design. In any practical situation, a source of finite,but small, extent can better be accommodated by the small-sourcenonimaging design described herein rather than by the existing prior artfinite-source designs.

We can idealize a source by a line or point with negligible diameter andseek a one-reflection solution in analogy with the conventional"edge-ray methods" of nonimaging optics (see, for example, W. T. Welfordand R. Winston "High Collection Nonimaging Optics," Academic Press, NewYork, N. Y. (1989)). Polar coordinates R, Φ are used with the source asorigin and θ for the angle of the reflected ray as shown in FIG. 3. Thegeometry in FIG. 3 shows that the following relation between sourceangle and reflected angle applies:

    d/dΦ(logR)=tanα,                                 (1)

where α is the angle of incidence with respect to the normal. Therefore,

    α=(Φ-θ)/2                                  (2)

Equation (1) is readily integrated to yield,

    log(R)=∫tan adΦ+const.                            (3)

so that,

    R=const. exp∫(tan adΦ)                            (4)

This equation (4) determines the reflector profile R(Φ) for any desiredfunctional dependence θ(Φ).

Suppose we wish to radiate power (P) with a particular angulardistribution P(θ) from a line source which we assume to be axiallysymmetric. For example, P(θ)=const. from θ=0 to θ₁ and P(θ)˜0 outsidethis angular range. By conservation of energy P(θ)dΦ=P(Φ)dΦ (neglectingmaterial reflection loss) we need only ensure that

    dθdΦ=P(Φ)/P(θ)                         (5)

to obtain the desire radiated beam profile. To illustrate the method,consider the above example of a constant P(θ) for a line source. Byrotational symmetry of the line source, dP/dΦ=a constant so that,according to Equation (4) we want θ to be a linear function of Φ suchas, θ=aΦ. Then the solution of Equation (3) is

    R=R.sub.0 /cos.sup.k (Φ/k)                             (6)

where,

    k-2/(1-a),                                                 (7)

and R₀ is the value of R at Φ=0. We note that the case a=0(k=2) givesthe parabola in polar form,

    R=R.sub.0 /cos.sup.2 (Φ/2),                            (8)

while the case θ=constant=θ₁ gives the off-axis parabola,

    R=R.sub.0 cos.sup.2 (θ.sub.1 /2)/cos.sup.2 [Φ-θ.sub.0)/2](9)

Suppose we desire instead to illuminate a plane with a particularintensity distribution. Then we correlate position on the plane withangle q and proceed as above. Turning next to a spherically symmetricpoint source, we consider the case of a constant P(Ω) where Ω is theradiated solid angle. Now we have by energy conservation,

    P(Ω)dΩ=P(Ω.sub.0)dΩ.sub.0          (10)

where Ω₀ is the solid angle radiated by the source. By sphericalsymmetry of the point source, P(Ω₀)=constant. Moreover, we have dΩ=(2π)dcosθ and dΩ₀ =(2π)d cosΦ; therefore, we need to make cosθ a linearfunction of cosΦ,

    cosθ=a cosΦ+b                                    (11).sub.1

With the boundary conditions that θ=0 at Φ=θ, θ=θ₁ at Φ=Φ₀, we obtain,

    a=(1-cosθ.sub.1)/(1-cosΦ.sub.0)                  (12)

    b=(cosθ.sub.1 -cosΦ.sub.0)/(1-cosΦ.sub.0)    (13)

[For example, for θ₁ <<1 and Φ₀ ˜π/2 we have, √2θ₀ sin(1/2Φ).] Thisfunctional dependence is applied to Equation (4) which is thenintegrated, such as by conventional numerical methods.

A useful way to describe the reflector profile R(Φ) is in terms of theenvelope (or caustic) of the reflected rays r(Φ). This is most simplygiven in terms of the direction of the reflected ray t=(-sinθ, cosθ).Since r(Φ) lies along a reflected ray, it has the form,

    r=R+Lt.                                                    (14)

where R=R(sinΦ₁ -cosΦ). Moreover,

    RdΦ=LdΦ                                            (15)

which is a consequence of the law of reflection. Therefore,

    r=R+t/(dθ/dΦ)                                    (16)

In the previously cited case where θ is the linear function aΦ, thecaustic curve is particularly simple,

    r=R+t/a                                                    (17)

In terms of the caustic, we may view the reflector profile R as thelocus of a taut string; the string unwraps from the caustic r while oneend is fixed at the origin.

In any practical design the small but finite size of the source willsmear by a small amount the "point-like" or "line-like" angulardistributions derived above. To prevent radiation from returning to thesource, one may wish to "begin" the solution in the vicinity of θ=0 withan involute to a virtual source. Thus, the reflector design should beinvolute to the "ice cream cone" virtual source. It is well known in theart how to execute this result (see, for example, R. Winston, "Appl.Optics," Vol. 17, p. 166 (1978)). Also, see U.S. Pat. No. 4,230,095which is incorporated by reference herein. Similarly, the finite size ofthe source may be better accommodated by considering rays from thesource to originate not from the center but from the periphery in themanner of the "edge rays" of nonimaging designs. This method can beimplemented and a profile calculated using the computer program of theAppendix (and see FIG. 2) and an example of a line source and profile isillustrated in FIG. 1. Also, in case the beam pattern and/or source isnot rotationally symmetric, one can use crossed two-dimensionalreflectors in analogy with conventional crossed parabolic shapedreflecting surfaces. In any case, the present methods are most usefulwhen the sources are small compared to the other parameters of theproblem.

Various practical optical sources can include a long arc source whichcan be approximated by an axially symmetric line source. We then canutilize the reflector profile R(Φ) determined hereinbefore as explainedin expressions (5) to (9) and the accompanying text. This analysisapplies equally to two and three dimensional reflecting surface profilesof the optical device.

Another practical optical source is a short arc source which can beapproximated by a spherically symmetric point source. The details ofdetermining the optical profile are shown in Equations (10) through(13).

A preferred form of nonimaging optical system 20 is shown in FIG. 4Awith a representative nonimaging output illustrated in FIG. 5A. Such anoutput can typically be obtained using conventional infrared opticalsources 22 (see FIG. 4A), for example high intensity arc lamps orgraphite glow bars. Reflecting side walls 24 and 26 collect the infraredradiation emitted from the optical source 22 and reflect the radiationinto the optical far field from the reflecting side walls 24 and 26. Anideal infrared generator concentrates the radiation from the opticalsource 22 within a particular angular range (typically a cone of about %15 degrees) or in an asymmetric field of % 20 degrees in the horizontalplane by % 6 degrees in the vertical plane. As shown from the contoursof FIG. 5B, the prior art paraboloidal reflector systems (not shown)provide a nonuniform intensity output, whereas the optical system 20provides a substantially uniform intensity output as shown in FIG. 5A.Note the excellent improvement in intensity profile from the prior artcompound parabolic concentrator (CPC) design. The improvements aresummarized in tabular form in Table I below:

                  TABLE I                                                         ______________________________________                                        Comparison of CPC with Improved Design                                                             CPC  New Design                                          ______________________________________                                        Ratio of Peak to On Axis Radiant Intensity                                                           1.58   1.09                                            Ratio of Azimuth Edge to On Axis                                                                     0.70   0.68                                            Ratio of Elevation Edge to On Axis                                                                   0.63   0.87                                            Ratio of Corner to On Axis                                                                           0.33   0.52                                            Percent of Radiation Inside Useful Angles                                                            0.80   0.78                                            Normalized Mouth Area  1.00   1.02                                            ______________________________________                                    

In a preferred embodiment designing an actual optical profile involvesspecification of four parameters. For example, in the case of aconcentrator design, these parameters are:

1. a=the radius of a circular absorber;

2. b=the size of the gap;

3. c=the constant of proportionality between θ and Φ-Φ₀ in the equationθ=c(Φ-Φ₀);

4. h=the maximum height.

A computer program has been used to carry out the calculations, andthese values are obtained from the user (see lines six and thirteen ofthe program which is attached as a computer software Appendix includedas part of the specification).

From Φ.sub. 0 to φ.sub.Φ= 0 in FIG. 3B the reflector profile is aninvolute of a circle with its distance of closest approach equal to b.The parametric equations for this curve are parameterized by the angle α(see FIG. 3A). As can be seen in FIG. 3B, as Φ varies from 0 to Φ₀, αvaries from α₀ to ninety degrees. The angle α₀ depends on a and b, andis calculated in line fourteen of the computer software program. Betweenlines fifteen and one hundred and one, fifty points of the involute arecalculated in polar coordinates by stepping through these parametricequations. The (r,θ) points are read to arrays r(i), and theta(i),respectively.

For values of Φ greater than Φ₀, the profile is the solution to thedifferential equation:

    d(lnr)dΦ=tan{ 1/2[Φ-θ+arc sin(a/r]}

where θ is a function of φ. This makes the profile r(φ) a functional ofθ. In the sample calculation performed, θ is taken to be a linearfunction of Φ as in step 4. Other functional forms are described in thespecification. It is desired to obtain one hundred fifty (r,theta)points in this region. In addition, the profile must be truncated tohave the maximum height, h. We do not know the (r, theta) point whichcorresponds to this height, and thus, we must solve the above equationby increasing phi beyond Φ₀ until the maximum height condition is met.This is carried out using the conventional fourth order Runga-Kuttanumerical integration method between lines one hundred two and onehundred and fifteen. The maximum height condition is checked betweenlines one hundred sixteen and one hundred twenty.

Once the (r, theta) point at the maximum height is known, we can set ourstep sizes to calculate exactly one hundred fifty (r, theta) pointsbetween φ₀ and the point of maximum height. This is done between linestwo hundred one and three hundred using the same numerical integrationprocedure. Again, the points are read into arrays r(i), theta(i).

In the end, we are left with two arrays: r(i) and theta(i), each withtwo hundred components specifying two hundred (r, theta) points of thereflector surface. These arrays can then be used for designspecifications, and ray trace applications.

In the case of a uniform beam design profile, (P(θ)=constant), a typicalset of parameters is (also see FIG. 1):

a=0.055 in.

b=0.100 in.

h=12.36 in.

c=0.05136

for θ(Φ)=c(Φ-Φ_(o))

In the case of an exponential beam profile (P(θ)=ce^(-a)θ) a typical setof parameters is: ##EQU1##

Power can be radiated with a particular angular distribution P°(θ) froma source which itself radiates with a power distribution P°(φ). Theangular characteristic of the source is the combined result of itsshape, surface brightness, and surface angular emissivity at each point.A distant observer viewing the source fitted with the reflector under anangle θ will see a reflected image of the source in addition to thesource itself. This image will be magnified by some factor [M] if thereflector is curved. Ideally both the source and its reflected imagehave the same brightness, so the power each produces is proportional tothe apparent size. The intensity perceived by the observer, P°(θ) willbe the sum of the two

    P°(θ)=P°(θ)+|M|P°(.theta.).                                                        (18)

The absolute value of the magnification has to be taken because, if thereflected image and the source are on different sides of the reflectorand we therefore perceive the image as reversed or upside down, then themagnification is negative. Actually, at small angles, the source and itsreflection image can be aligned so that the observer perceives only thelarger of the two. But if [M] is large, one can neglect the directradiation from the source.

Thus, one is concerned with the magnification of the reflector. Adistant observer will see a thin source placed in the axis of a troughreflector magnified in width by a factor ##EQU2##

This can be proved from energy conservation since the power emitted bythe source is conserved upon reflection: P^(Sd)φ=MP^(S) dθ.

For a rotationally symmetric reflector the magnification, Mm as given inEq.(19) refers to the meridional direction. In the sagittal directionthe magnification is ##EQU3## where now μ₁ and μ₂ are small angles inthe sagittal plane, perpendicular to the cross section shown in FIG. 2.Equation (20) can be easily verified by noting that the sagittal imageof an object on the optical axis must also lie on the optical axis. Thereason for this is that because of symmetry, all reflected rays must becoplanar with the optical axis.

The total magnification M_(t) is the product of the sagittal and themeridional magnification ##EQU4##

Actually Eq.(21) could also have been derived directly from energyconservation by noting that the differential solid angle is proportionalto d cos(θ) and d cos(φ) respectively.

Thus, inserting the magnification given in Eq.(21) or Eq.(19), as thecase may be, into Eq.(18) yields the relationship between φ and θ whichproduces a desired power distribution P(θ) for a given angular powerdistribution of the source P^(S). This relationship then can beintegrated as outlined in Eq. 17 to construct the shape of the reflectorwhich solves that particular problem

There are two qualitatively different solutions depending on whether weassume the magnification to be positive or negative. If Mm>0 this leadsto CEC-type devices, whereas Mtn<0 leads to CHC-type devices. The termCEC refers to Compound Elliptical Concentrator and CHC to the so calledCompound Hyperbolic Concentrator.

Now the question arises of how long we can extend the reflector or overwhat angular range we can specify the power distribution. From Eq.(17)one can see that if φ-θ=π then R diverges. In the case of negativemagnification this happens when the total power seen by the observerbetween θ=0 and θ=θ^(max) approaches the total power radiated by thesource between φ=0 and φ=π. A similar limit applies to the opposite sideand specifies θ^(min). The reflector asymptotically approaches aninfinite cone or V-trough. There is no power radiated or reflectedoutside the range θ^(min) <θ<θ^(max).

For positive magnification the reflected image is on the opposite sideof the symmetry axis (plane) to the observer. In this case the limit ofthe reflector is reached as the reflector on the side of the observerstarts to block the source and its reflection image. For symmetricdevices this happens when φ+θ=π. In this case too one can show that thelimit is actually imposed by the first law. However, the reflectorremains finite in this limit. It always ends with a vertical tangent.For symmetric devices where θ^(max) =-θ^(min) and φ^(max) --φ^(min) theextreme directions for both the CEC-type and the CHC-type solution arerelated by

    φ.sup.max +θ.sup.max =π                       (22)

In general CEC-type devices tend to be more compact. The mirror areaneeded to reflect a certain beam of light is proportional to 1/cos(α).The functional dependence of θ and φ for symmetrical problems is similarexcept that they have opposite signs for CHC-type devices and equalsigns for CEC-type solutions. Therefore α increases much more rapidlyfor the CHC-type solution which therefore requires a larger reflector,assuming the same initial value R₀. This is visualized in FIG. 8 andwhere the acceptance angle function as well as the incidence angle α areboth plotted for the negative magnification solution.

To illustrate rate the above principles, consider a strip source as anexample. For a narrow, one-sided lambertian strip, the radiant power inproportional to the cosine of the angle. In order to produce a constantirradiance on a distant target the total radiation of source andreflection should therefore be proportional to 1/cos² (θ). This yields##EQU5##

The boundary condition is, in this case, θ=0 at φ=±π/2 because we assumethat the strip only radiates on one side, downward. Equation 11 can onlybe integrated for α=1:

    sin(φ)=1-|tan(θ)-sin(θ)|.(24)

The acceptance angle function θ as well as the incidence angle for theCEC-type solution are shown in FIG. 8. Integrating yields the reflectorshapes plotted in FIG. 9.

The analytical tools described herein can be used to solve real problemswhich involve reflectors close to the source. This is done by combiningthe above technique with the edge ray method which has proved soeffective in nonimaging designs. That is, the above methods can beapplied to edge rays. As a first example, a reflector is designed for aplanar, lambertian strip source so as to achieve a predeterminedfar-field irradiance. The reflector is designed so that the reflectedimage is immediately adjacent to the source. This is only possible in anegative magnification arrangement. Then the combination of source andits mirror image is bounded by two edge rays as indicated in FIG. 10.The combined angular power density for a source of unit brightnessradiated into a certain direction is given by the edge ray separation.

    R sin(2α)=P°(θ).

By taking the logarithmic derivative of Eq.(25) and substituting thefollowing: ##EQU6##

This describes the right hand side, where θ<0. The other side is themirror image.

For 2α=π, R diverges just as in the case of the CHC-type solutions forsmall sources. Thus, in general the full reflector extends to infinity.For practical reasons it will have to be truncated. Let's assume thatthe reflector is truncated at a point T from which the edge ray isreflected into the direction θτ. For angles θ in between ±θτ thetruncation has no effect because the outer parts of the reflector do notcontribute radiation in that range. Therefore within this range thetruncated reflector also produces strictly the desired illumination.Outside this range the combination of source plus reflector behaves likea flat source bounded by the point T and the opposite edge of thesource. Its angular power density is given by Eq.(13) withR=Rτ=constant. The total power Pτ radiated beyond θτ is thus ##EQU7##

In order to produce an intensity P°(θτ) at θτ, R(θτ) must be ##EQU8##The conservation of total energy implies that the truncated reflectorradiates the same total power beyond θτ as does the untruncatedreflector. ##EQU9## This equation must hold true for any truncationθ=θτ. It allows us to explicitly calculate α, and with it φ and R, inclosed form as functions of θ, if B(θ), that is the integral of P°(θ) isgiven in closed form. The conservation of total energy also implies thatthe untruncated reflector radiates the same total power as the baresource. This leads to the normalizing condition. ##EQU10##

This condition may be used to find θ^(max) ; it is equivalent to settingθτ=0, 2ατ=π/2 in Eq.(30). Solving Eq.(30) for α yields. ##EQU11##

Substituting α=(φ-θ)12, yields the acceptance angle function

    φ(θ)=θ+2α.                           (33)

From Eq.(25) the radius is given by ##EQU12##

These equations specify the shape of the reflector in a parametric polarrepresentation for any desired angular power distribution P°(θ). Astraight forward calculation shows that Eq.(32) is indeed the solutionof the differential equation (27). In fact, Eq.(27) was not needed forthis derivation of the reflector shape. We have presented it only toshow the consistency of the approach.

For example, to produce a constant irradiance on a plane parallel to thesource we must have P°(θ)=1/cos² (θ) and thus B(θ)=cos²(θ)(tan(θ)-tan(θmax)). Using Eq.(31), we find θmax=-π/4 so thatB(θ)=cos² (θ)(tan(θ)+1) with no undetermined constants.

The resulting acceptance angle function and the reflector profile areshown in FIG. 11 and FIG. 17 respectively. The reflector shape is closeto a V-trough. Though, the acceptance angle function is only poorlyapproximated by a straight line, which characterizes the V-trough. InFIG. 13 we show the deviation of the reflector shape depicted in FIG. 12from a true V-trough. Note, that a true V-trough produces a markedlynon-constant irradiance distribution proportional to cos(θ+π/4)cos(θ)for 0<-θ<π/4.

As a second example for a specific non-constant irradiance a reflectorproduces the irradiance distribution on a plane shown in FIG. 14. Thecorresponding angular power distribution is shown in FIG. 15. Theacceptance angle function according to Eq.(33) and (32) and theresulting reflector shape according to Eq.(34) are visualized in FIG. 16and FIG. 17.

Although the desired irradiance in this example is significantlydifferent from the constant irradiance treated in the example before,the reflector shape again superficially resembles a V-trough and thereflector of the previous example. The subtle difference between thereflector shape of this example and a true V-trough are visualized inFIG. 18 and FIG. 19 where we plot the slope of our reflector and thedistance to a true V-trough. Most structure is confined to the regionadjacent to the source. The fact that subtle variations in reflectorshape have marked effects on the power and irradiance distribution ofthe device can be attributed to the large incidence angle with which theedge rays strike the outer parts of the reflector.

As mentioned before, in general the reflector is of infinite size.Truncation alters, however, only the distribution in the outer parts. Toillustrate the effects of truncation for the reflector of this example,we plot in FIG. 20 the angle up to which the truncated device matchesthe desired power distribution, as a function of the vertical length ofthe reflector. Thus for example the truncated device shown in FIG. 17has the irradiance distribution and power distribution shown in brokenline in FIG. 14 and FIG. 15. Note that the reflector truncated to avertical length of 3 times the source width covers more than 5/6 of theangular range.

B. General Optical Sources

Nonimaging illumination can also be provided by general optical sourcesprovided the geometrical constraints on a reflector can be defined bysimultaneously solving a pair of system. The previously recitedequations (1) and (2) relate the source angle and angle of lightreflection from a reflector surface,

    d/dφ(logR.sub.i)=tan (φ.sub.i -θ.sub.i)/2

and the second general expression of far field illuminance is,

    L(θ.sub.i)·R.sub.i sin(φ.sub.i -θ.sub.i)G(θ.sub.i)=I(θ.sub.i)

where L (θ_(i)) is the characteristic luminance at angle θ_(i) and G(θ_(i)) is a geometrical factor which is a function of the geometry ofthe light source. In the case of a two dimensional Lambertian lightsource as illustrated in FIG. 6A, the throughput versus angle forconstant illuminance varies as cos² θ. For a three dimensionalLambertian light source shown in FIG. 6D, the throughput versus anglefor constant illuminance varies as cos³ θ.

Considering the example of a two dimensional Lambertian light source andthe planar source illustrated in FIG. 6B, the concept of using a generallight source to produce a selected far field illuminance can readily beillustrated. Notice with our sign convention angle θ in FIG. 6B isnegative. In this example we will solve equations (18) and (19)simultaneously for a uniform far field illuminance using the twodimensional Lambertian source. In this example, equation (19) because,

    R.sub.i sin(φ.sub.i -θ.sub.i)cos.sup.2 θ.sub.i =I(θ.sub.i)

Generally for a bare two dimensional Lambertian source,

    I(θ.sub.i)˜δcosθ.sub.i

    δ˜a cosθ.sub.i /l

    l˜d/cos θ

Therefore, I˜cos³ θ.

In the case of selecting a uniform far field illumanance I(θ_(i))=C, andif we solve the equations at the end of the first paragraph of sectionB.,

    d/dφ(log R.sub.i)=tan (φ.sub.i -θ.sub.i)/2 and

    log R.sub.i +log sin(φ.sub.i -θ.sub.i)+2log cosθ.sub.i =logC=constant

    solving dφ.sub.i /dθ.sub.i =-2tanθ.sub.i sin(φ.sub.i -θ.sub.i)-cos(φ.sub.i -θ.sub.i)

    or let Ψ.sub.i =φ.sub.i -θ.sub.i

    dΨ.sub.i /dθ.sub.i =1+sin Ψ.sub.i -2tanθ.sub.i cos Ψ.sub.i

Solving numerically by conventional methods, such as the Runge-Kuttamethod, starting at Ψ_(i) =0 at θ_(i), for the constant illuminance,

dΨ_(i) /dθ_(i) =1+sinΨ_(i) -n tan θ_(i) cos Ψ_(i) where n is two for thetwo dimensional source. The resulting reflector profile for the twodimensional solution is shown in FIG. 6C and the tabulated datacharacteristic of FIG. 6C is shown in Table III. The substantially exactnature of the two dimensional solution is clearly shown in the ray tracefit of FIG. 7A. The computer program used to perform these selectivecalculation is included as Appendix B of the Specification. For a barethree dimensional Lambertian source where I(θ_(i))˜cos⁴ θ_(i), n islarger than 2 but less than 3.

The ray trace fit for this three dimensional solution is shown in FIG.7B wherein the "n" value was fitted for desired end result of uniformfar field illuminance with the best fit being about n=2.1.

Other general examples for different illuminance sources include,

(1) I(θ_(i))=A exp (Bθ_(i)) for a two dimensional, exponentialilluminance for which one must solve the equation,

    dΨ.sub.i /dθ.sub.i =1+sinΨ.sub.i -2tan θ.sub.i cosΨ+B

(2) I(θ_(i))=A exp (-Bθ_(i) ² /2) for a two dimensional solution for aGaussian illuminance for which one must solve,

    dΨ.sub.i /dθ.sub.i =1+sinΨ.sub.i -2tanθ.sub.i cosΨ.sub.i -Bθ.sub.i

The equations in the first paragraph of section B can of course begeneralized to include any light source for any desired for fieldilluminance for which one of ordinary skill in the art would be able toobtain convergent solutions in a conventional manner.

A ray trace of the uniform beam profile for the optical device of FIG. 1is shown in a tabular form of output in Table II below:

                  TABLE II                                                        ______________________________________                                        114 202 309 368 422 434 424 608 457 448 400 402 315 229 103                   145 295 398 455 490 576 615 699 559 568 511 478 389 298 126                   153 334 386 465 515 572 552 622 597 571 540 479 396 306 190                   202 352 393 452 502 521 544 616 629 486 520 432 423 352 230                   197 362 409 496 496 514 576 511 549 508 476 432 455 335 201                   241 377 419 438 489 480 557 567 494 474 482 459 421 379 230                   251 364 434 444 487 550 503 558 567 514 500 438 426 358 231                   243 376 441 436 510 526 520 540 540 482 506 429 447 378 234                   233 389 452 430 489 519 541 547 517 500 476 427 442 344 230                   228 369 416 490 522 501 539 546 527 481 499 431 416 347 227                   224 359 424 466 493 560 575 553 521 527 526 413 417 320 205                   181 378 392 489 485 504 603 583 563 530 512 422 358 308 194                   150 326 407 435 506 567 602 648 581 535 491 453 414 324 179                   135 265 382 450 541 611 567 654 611 522 568 446 389 300 130                   129 213 295 364 396 404 420 557 469 435 447 351 287 206 146                     ELEVATION                                                                   ______________________________________                                    

                  TABLE III                                                       ______________________________________                                        Phi           Theta    r                                                      ______________________________________                                        90.0000       0.000000 1.00526                                                90.3015       0.298447 1.01061                                                90.6030       0.593856 1.01604                                                90.9045       0.886328 1.02156                                                91.2060       1.17596  1.02717                                                91.5075       1.46284  1.03286                                                91.8090       1.74706  1.03865                                                92.1106       2.02870  1.04453                                                92.4121       2.30784  1.05050                                                92.7136       2.58456  1.05657                                                93.0151       2.85894  1.06273                                                93.3166       3.13105  1.06899                                                93.6181       3.40095  1.07536                                                93.9196       3.66872  1.08182                                                94.2211       3.93441  1.08840                                                94.5226       4.19810  1.09507                                                94.8241       4.45983  1.10186                                                95.1256       4.71967  1.10876                                                95.4271       4.97767  1.11576                                                95.7286       5.23389  1.12289                                                96.0302       5.48838  1.13013                                                96.3317       5.74120  1.13749                                                96.6332       5.99238  1.14497                                                96.9347       6.24197  1.15258                                                97.2362       6.49004  1.16031                                                97.5377       6.73661  1.16817                                                97.8392       6.98173  1.17617                                                98.1407       7.22545  1.18430                                                98.4422       7.46780  1.19256                                                98.7437       7.70883  1.20097                                                99.0452       7.94857  1.20952                                                99.3467       8.18707  1.21822                                                99.6482       8.42436  1.22707                                                99.9498       8.66048  1.23607                                                100.251       8.89545  1.24522                                                100.553       9.12933  1.25454                                                100.854       9.36213  1.26402                                                101.156       9.59390  1.27367                                                101.457       9.82466  1.28349                                                101.759       10.0545  1.29349                                                102.060       10.2833  1.30366                                                102.362       10.5112  1.31402                                                102.663       10.7383  1.32457                                                102.965       10.9645  1.33530                                                103.266       11.1899  1.34624                                                103.568       11.4145  1.35738                                                103.869       11.6383  1.36873                                                104.171       11.8614  1.38028                                                104.472       12.0837  1.39206                                                104.774       12.3054  1.40406                                                105.075       12.5264  1.41629                                                105.377       12.7468  1.42875                                                105.678       12.9665  1.44145                                                105.980       13.1857  1.45441                                                106.281       13.4043  1.46761                                                                       1.48108                                                107.789       14.4898  1.53770                                                108.090       14.7056  1.55259                                                108.392       14.9209  1.56778                                                108.693       15.1359  1.58329                                                108.995       15.3506  1.59912                                                109.296       15.5649  1.61529                                                109.598       15.7788  1.63181                                                109.899       15.9926  1.64868                                                110.201       16.2060  1.66591                                                110.503       16.4192  1.68353                                                110.804       16.6322  1.70153                                                111.106       16.8450  1.71994                                                111.407       17.0576  1.73876                                                111.709       17.2701  1.75801                                                112.010       17.4824  1.77770                                                112.312       17.6946  1.79784                                                112.613       17.9068  1.81846                                                112.915       18.1188  1.83956                                                113.216       18.3309  1.86117                                                113.518       18.5429  1.88330                                                113.819       18.7549  1.90596                                                114.121       18.9670  1.92919                                                114.422       19.1790  1.95299                                                114.724       19.3912  1.97738                                                115.025       19.6034  2.00240                                                115.327       19.8158  2.02806                                                115.628       20.0283  2.05438                                                115.930       20.2410  2.08140                                                116.231       20.4538  2.10913                                                116.533       20.6669  2.13761                                                116.834       20.8802  2.16686                                                117.136       21.0938  2.19692                                                117.437       21.3076  2.22782                                                117.739       21.5218  2.25959                                                118.040       21.7362  2.29226                                                118.342       21.9511  2.32588                                                118.643       22.1663  2.36049                                                118.945       22.3820  2.39612                                                119.246       22.5981  2.43283                                                119.548       22.8146  2.47066                                                119.849       23.0317  2.50967                                                120.151       23.2493  2.54989                                                120.452       23.4674  2.59140                                                120.754       23.6861  2.63426                                                121.055       23.9055  2.67852                                                121.357       24.1255  2.72426                                                121.658       24.3462  2.77155                                                121.960       24.5676  2.82046                                                122.261       24.7898  2.87109                                                122.563       25.0127  2.92352                                                122.864       25.2365  2.97785                                                123.166       25.4611  3.03417                                                123.467       25.6866  3.09261                                                123.769       25.9131  3.15328                                                124.070       26.1406  3.21631                                                124.372       26.3691  3.28183                                                124.673       26.5986  3.34999                                                124.975       26.8293  3.42097                                                125.276       27.0611  3.49492                                                125.578       27.2941  3.57205                                                125.879       27.5284  3.65255                                                126.181       27.7640  3.73666                                                126.482       28.0010  3.82462                                                126.784       28.2394  3.91669                                                127.085       28.4793  4.01318                                                127.387       28.7208  4.11439                                                127.688       28.9638  4.22071                                                127.990       29.2086  4.33250                                                128.291       29.4551  4.45022                                                128.593       29.7034  4.57434                                                128.894       29.9536  4.70540                                                129.196       30.2059  4.84400                                                129.497       30.4602  4.99082                                                129.799       30.7166  5.14662                                                130.101       30.9753  5.31223                                                130.402       31.2365  5.48865                                                130.704       31.5000  5.67695                                                131.005       31.7662  5.87841                                                131.307       32.0351  6.09446                                                131.608       32.3068  6.32678                                                131.910       32.5815  6.57729                                                132.211       32.8593  6.84827                                                132.513       33.1405  7.14236                                                132.814       33.4251  7.46272                                                133.116       33.7133  7.81311                                                133.417       34.0054  8.19804                                                133.719       34.3015  8.62303                                                134.020       34.6019  9.09483                                                134.322       34.9068  9.62185                                                134.623       35.2165  10.2147                                                134.925       35.5314  10.8869                                                135.226       35.8517  11.6561                                                135.528       36.1777  12.5458                                                135.829       36.5100  13.5877                                                136.131       36.8489  14.8263                                                136.432       37.1949  16.3258                                                136.734       37.5486  18.1823                                                137.035       37.9106  20.5479                                                137.337       38.2816  23.6778                                                137.638       38.6625  28.0400                                                137.940       39.0541  34.5999                                                138.241       39.4575  45.7493                                                138.543       39.8741  69.6401                                                138.844       40.3052  166.255                                                139.146       40.7528  0.707177E-01                                           139.447       41.2190  0.336171E-01                                           139.749       41.7065  0.231080E-01                                           140.050       42.2188  0.180268E-01                                           140.352       42.7602  0.149969E-01                                           140.653       43.3369  0.129737E-01                                           140.955       43.9570  0.115240E-01                                           141.256       44.6325  0.104348E-01                                           141.558       45.3823  0.958897E-02                                           141.859       46.2390  0.891727E-02                                           142.161       47.2696  0.837711E-02                                           142.462       48.6680  0.794451E-02                                           142.764       50.0816  0.758754E-02                                           143.065       48.3934  0.720659E-02                                           143.367       51.5651  0.692710E-02                                           143.668       51.8064  0.666772E-02                                           143.970       56.1867  0.647559E-02                                           144.271       55.4713  0.628510E-02                                           144.573       54.6692  0.609541E-02                                           144.874       53.7388  0.590526E-02                                           145.176       52.5882  0.571231E-02                                           145.477       50.8865  0.550987E-02                                           145.779       53.2187  0.534145E-02                                           146.080       52.1367  0.517122E-02                                           146.382       50.6650  0.499521E-02                                           146.683       49.5225  0.481649E-02                                           146.985       45.6312  0.459624E-02                                           147.286       56.2858  0.448306E-02                                           147.588       55.8215  0.437190E-02                                           147.889       55.3389  0.426265E-02                                           148.191       54.8358  0.415518E-02                                           148.492       54.3093  0.404938E-02                                           148.794       53.7560  0.394512E-02                                           149.095       53.1715  0.384224E-02                                           149.397       52.5498  0.374057E-02                                           149.698       51.8829  0.363992E-02                                           150.000       51.1597  0.354001E-02                                           ______________________________________                                    

C. Extended, Finite Size Sources

In this section we demonstrate how compact CEC-type reflectors can bedesigned to produce a desired irradiance distribution on a given targetspace from a given, finite size, source. The method is based ontailoring the reflector to a family of edge-rays, but at the same timethe edge rays of the reflected source image are also controlled.

In order to tailor edge rays in two dimensions, for example, one canassume a family of edge rays, such as are produced by a luminancesource. Through each point in the space outside the luminance sourcethere is precisely one edge ray. The direction of the edge rays is acontinuous and differentiable (vector) function of position. If we havea second, tentative family of edge-rays represented by anothercontinuous vector function in the same region of space, we can design areflector which precisely reflects one family onto the other. Each pointin space is the intersection of precisely one member of each family.Therefore, the inclination of the desired reflector in each point inspace can be calculated in a conventional, well known manner. Thus, onecan derive a differential equation which uniquely specifies thereflector once the starting point is chosen.

We can, for example, formalize this idea for the case where thetentative family of edge rays is given only along a reference line whichis not necessarily a straight line. This pertains to the usual problemsencountered in solving illumination requirements.

Referring to FIG. 21, let a=a(x) be the two dimensional unit vector 100pointing towards the edge of a source 102 as seen from a point x, wherek=k(t) is a parameterization of reference line 104 according to a scalarparameter t. Let u(t) be a unit vector 106 pointing in the direction ofan edge ray 107 desired at the reference location specified by t. We canparametrize the contour of a reflector 108 with respect to the referenceline 104 by writing the points on the reflector 108 as:

    R(t)=k(t)+Du(t)                                            (35)

Here the scalar D denotes the distance from a point on the referenceline 104 to the reflector 108 along the desired edge ray 107 throughthis point.

Designing the shape of the reflector 108 in this notation is equivalentto specifying the scalar function D=D(t). An equation for D is derivedfrom the condition that the reflector 108 should reflect the desirededge ray 107 along u(t) into the actual edge ray a(R(t)) and vice versa:##EQU13## Inserting Eq. (35) from above yields: ##EQU14##

Here the dots indicate scalar products. Equation (37) is a scalardifferential equation for the scalar function D(t). By solving thisequation, we can determine the reflector 108 which tailors the desiredfamily of the edge ray 107 specified by the unit vector, u, 106 to thesource 102 characterized by the vector function, a.

This approach can also be used to tailor one family of the edge rays 107onto another with refractive materials rather then reflectors. Equation(36) then is replaced by Snell's law.

Consequently, the condition for the existence of a solution in thisembodiment is that each point on the reflector 108 is intersected byprecisely one member of the family of tentative edge rays. To be able todefine this family of edge rays 107 along the reference line 104, eachpoint on the reference line 104 must also be intersected by preciselyone tentative edge ray. This is less than the requirement that thetentative edge rays define a physical surface which produces them. Thefamily of the edge rays 107 of a physical contour (for example rightedge rays) must also satisfy, the further requirement that precisely oneedge ray passes through each point of the entire space exterior to thecontour. Indeed we can produce families of such edge rays by tailoring,but which cannot be produced by a single physical source. This isconfirmed by observations that curved mirrors produce not only adistorted image of the source, but furthermore an image is produced thatappears to move as the observer moves.

The condition that each point on the reflector 108, as well as eachpoint on the reference line 104, should be intersected by precisely oneof the desired edge rays 107 implies that the caustic formed by theseedge rays 107 cannot intersect the reflector 108 or the reference line104. The caustic must therefore either be entirely confined to theregion between the reflector 108 and the reference line 104, or lieentirely outside this region. The first of these alternativescharacterizes the CEC-type solutions, while the second one definesCHC-type solutions.

In order to determine the desired edge rays 107, the irradiance, forexample, from a Lambertian source of uniform brightness B is given byits projected solid angle, or view factor. In a conventional, knownmanner the view factor is calculated by projecting the source 102 firston a unit sphere surrounding the observer (this yields the solid angle)and then projecting the source 102 again onto the unit circle tangent tothe reference plane. The view factor is determined by the contour of thesource 102 as seen by the observer. In two dimensions for example, theirradiance E is,

E=B(sinr_(R) -sinr_(L))

where r_(R) and r_(L) are the angles between the normal to the referenceline and the right and left edge rays striking the observer,respectively. If we know the brightness B, the desired irradiance E andone edge ray, then Eq. (38) can be used to determine the desireddirection of the other edge ray.

Consider the example of a source 110 of given shape (see FIG. 22). Wethen know the direction of the edge rays as seen by an observer as afunction of the location of the observer. The shape of the source 110can be defined by all its tangents. We can now design the reflector 108so that it reflects a specified irradiance distribution onto the givenreference line 104 iteratively.

In this iterative process if an observer proceeds, for example, fromright to left along reference line 112, the perceived reflection movesin the opposite direction for a CEC-type solution. As shown in FIG. 22Aa right edge ray 114 as seen by the observer, is the reflection of theright edge, as seen from reflector 116, and further plays the role ofleading edge ray 114' along the reflector 116. A left edge ray 118 isjust trailing behind, and this is shown in FIG. 22A as reflectedtrailing edge ray 118'. For a CHC-type reflector 126 (see FIG. 22B) thereflected image of the source 110 moves in the same direction as theobserver, and the right edge as seen by the observer is the reflectionof the left edge. If part of the reflector 126 is known, then a trailingedge ray 128' which has been reflected by the known part of thereflector 126, can be calculated as a function of location on thereference line 112. Equation (38) consequently specifies the directionof leading edge ray 130. Then, Eq. (37) can be solved to tailor the nextpart of the reflector profile to this leading edge ray 130. Consideringthe boundary conditions, if the reflector 116 or 126 is terminated, thenthe reflected radiation does not terminate where the leading edge fromthe end of the reflector 116 or 126 strikes the reference line 112.Rather the reflected radiation ends where the trailing edge from the endof the reflector 116 or 126 strikes the reference line 112 (see FIG.23). Thus, there is a `decay` zone 13 1 on the reference line 112 whichsubtends an equal angle at the source 110 as seen from the end of thereflector 116 or 126. In this region the previously leading edge is atan end location 129 of the reflector 116 or 126, while the trailing edgegradually closes in. An analogous `rise` zone 132 exists at the otherend of the reflector 116, 126, where the trailing edge is initiallyfixed to a `start` position 134 of the reflector 116. However, there isan important conceptual difference between these two regions, in thatthe `rise` of the irradiance can be modeled by tailoring the reflector116, 126 to the leading edge, while the `decay` cannot be influencedonce the reflector 116, 126 is terminated. Therefore, there is adifference in which way we can proceed in the iterative tailoring of thereflector 116, 126.

If the source 110 radiates in all directions and we want to avoidtrapped radiation (that is radiation reflected back onto the source110), then the reflected radiation from each side of the reflector 140should cover the whole target domain of the reflector 140 (see FIG. 24).At the same time, the normal to the reflector surface should notintersect the source 110. Therefore, lea and right side portions 142 and143, respectively, of the reflectors 140 are joined in a cusp. Anobserver in the target domain thus perceives radiation from two distinctreflections of the source 110, one in each of the portions 142 and 143of the reflector 140, in addition to the direct radiation from thesource 110.

If we assume symmetry as shown in FIG. 24 and that the reflector 140 ispreferred to be continuous and differentiable (except for the cusp inthe symmetry plane), then we require that as seen from the symmetryplane, the two perceived reflections are equal. For all other points inthe target domain we now have the additional degree of freedom ofchoosing the relative contributions of each of the portions 142 and 143of the reflector 140. In CEC-type solutions both reflections appear tobe situated between the target space and the reflector 140. Thus, as theobserver moves, both reflection images move in the opposite direction.To that end, when the observer approaches the outermost part of theilluminated target region, the reflection on the same side firstdisappears at the cusp in the center. Thereafter, the reflectionopposite to the observer starts to disappear past the outer edge of theopposite reflector, while the source itself is shaded by the outer edgeof the other reflector portion on the observer side. These eventsdetermine the end point of the reflector 140 because now the totalradiation in the target region equals the total radiation emitted by thesource 110.

CEC-Type Reflector for Constant Irradiance

A CEC-type reflector 150 can produce a constant irradiance on a distantplane from a finite size cylindric source 152 of uniform brightness.This requires the angular power distribution to be proportional to1/cos² (θ). In FIG. 25 we show the necessary power from both reflectionsso that the total power is as required. The reflector 150 is depicted inFIG. 26. The reflector 150 is designed starting from cusp 154 in thesymmetry axes. Note that each reflection irradiates mostly the oppositeside, but is visible from the same side too. Some angles have beenparticularly designated by the letters A through and E in FIG. 25. Thecorresponding edge rays are indicated also in FIG. 27.

Between -A and A angles the reflections are immediately adjacent to thesource 152. The cusp 154 in the center is not visible. Between A and Bangles the reflection from the same side as the observer slowlydisappears at the cusp 154, while the other increases in size forcompensation. Starting with C the source 152 is gradually eclipsed bythe end of the reflector 150. The largest angle for which a constantirradiance can be achieved is labeled D. The source 152 is not visible.The power is produced exclusively by the opposite side reflection. Thereflector 152 is truncated so that between D and E the reflectiongradually disappears at the end of the reflector 152.

The inner part of the reflector 150 which irradiates the same side, issomewhat arbitrary. In the example shown, we have designed it as aninvolute because this avoids trapped radiation and at the same timeyields the most compact design. At the center the power from eachreflection is very nearly equal to that of the source 152 itself. Oncethe power radiated to the same side is determined, the reflector 150 isdesigned so that the sum of the contributions of the two reflections andthe source 152 matches the desired distribution. Proceeding outward, theeclipsing of the source 152 by the reflector 150 is not known at first,because it depends on the end point. This problem is solved by iteratingthe whole design procedure several times.

The point of truncation is determined by the criterium that thereflector 150 intersects the edge rays marked B from the cusp 154. Thisis because the preferred design is based on a maximum of one reflection.This criterium is also the reason for designing the inner part as aninvolute.

The angular decay range D to E in FIGS. 25 and 27 depends only on thedistance of the end point to the source 152. Depending on the startingdistance from the cusp 154 to the source 152, the device can be designedeither more compact, but with a broader decay zone, or larger, and witha more narrow decay zone. The reflector 150 shown has a cusp distance of2.85 source diameters. The end point is at a distance of 8.5 sourcediameters. This ensures that a constant irradiance is produced between-43 and 43 degrees. The decay zone is only 7 degrees. This design waschosen in order that the source 152 is eclipsed just before the angle oftruncation.

The reflector 150 cannot be made much more compact as long as onedesigns for a minimum of one reflection. At the angle D the opening isnearly totally filled with radiation as seen in FIG. 27. The distancethe reflector 150 extends downward from the source 152 is alsodetermined by the maximum power required to produce at angle D. Thedistance of the cusp 154 also cannot be diminished, otherwise thecriterium for the end of the reflection 150 is reached sooner, thereflector 150 has to be truncated and the maximum power produced is alsoless.

The embodiments described hereinbefore involve at most one reflection.However, in other forms of the invention various systems based onmultiple reflections can be designed using the teachings providedherein. As more reflections contribute, the freedom of the designerincreases. This freedom can be used to adapt the reflector to othercriteria, such as a need for compactness. In any case, independent ofthe number of reflections, once the general architecture has beendetermined, tailoring the reflector to one set of edge rays determinesits shape without the need for approximations or a need to undergooptimizations.

We claim:
 1. A nonimaging illumination optical device for producing aselected far field illuminance I output over an angular range,comprising:a source of light having a characteristic irradiance, a lightreflecting surface having a reflector contour and a family of edge raysdefined along a reference line, a point located on said reference lineand a desired edge ray passing through the point and extending to saidreflecting surface, said reflector contour defined in terms of saidreference line by the expression R(t) where t is a scalar parameterposition:

    R(t)=k(t)+Du(t)

where k(t)=a parametrization of said reference line; D=a distance fromthe point on said reference line to said reflecting surface along saiddesired edge ray through the point; u=unit vector along the desireddirection of said desired edge ray; and said reflector contourreflecting said desired edge ray into actual edge ray a (R(t)), where ais a multidimensional unit vector pointing towards an edge of saidsource of light, and said D being the solutions of: ##EQU15##
 2. Theoptical device as defined in claim 1 wherein a caustic formed by saiddesired edge ray is confined to a region between said reflecting surfaceand said reference line.
 3. The optical device as defined in claim 1wherein a caustic formed by said desired edge ray is entirely outside aregion between said reflecting surface and said reference line.
 4. Theoptical device as defined in claim 1 wherein said irradiance is afunction additionally of brightness and said R for the angles betweenthe normal to said reference line and right and left edge rays strikingan observer, said function of irradiance usable to iteratively tailorsaid reflector contour.
 5. The optical device as defined in claim 4wherein said source of light is defined by tangent lines and a beginningknown surface contour of said reflector contour generates a trailingedge ray which can be used in conjunction with said irradiance functionto calculate a leading edge ray.
 6. The optical device as defined inclaim 1 wherein said reflecting surface comprises at least two reflectorcontours having a cusp therebetween.
 7. The optical device as defined inclaim 6 wherein said two reflector have a mirror image surface aboutsaid cusp.
 8. The optical device as defined in claim 1 wherein dR(t)/dtis perpendicular to a(R(t)-u(t)).
 9. The optical device as defined inclaim 1 wherein said reflector contour is comprised of a compoundelliptical concentrator reflector thereby causing a perceived reflectionto move in a direction opposite to a direction an observer moves. 10.The optical device as defined in claim 1 wherein said reflector contouris comprised of a compound hyperbolic concentrator reflector therebycausing a perceived reflection to move in a direction an observer moves.11. The optical device as defined in claim 1 wherein the actual edge rayand the desired edge ray comprise an edge ray which has undergone atmost a single reflection and an edge ray which has undergone multiplereflections, respectively.
 12. An optical device for collecting light ofirradiance I from a radiant source present over an output angular range,comprising:a transducer collecting device for light; a light reflectingsurface having a reflector contour and a family of edge rays definedalong a reference line, a point on said reference line and a desirededge ray passing through the point and extending to said reflectingsurface, said reflector contour defined in terms of said reference lineby an expression R(t) where t is a scalar parameter position, and:

    R(t)=k(t)+Du(t)

where k(t)=a parametrization of said reference line, D=a distance fromthe point on said reference line to said reflecting surface along thedesired edge ray through the point; u=a unit vector along the desireddirection of said desired edge ray; and said reflector contourreflecting said desired edge ray into actual edge ray, a (R(t)), where ais a multidimensional unit vector pointing towards an edge of saidtransducer of light and said D being the solutions of: ##EQU16##
 13. Theoptical device as defined in claim 12 wherein a caustic formed by saiddesired edge ray is confined to a region between said reflecting surfaceand said reference line.
 14. The optical device as defined in claim 12wherein a caustic formed by said desired edge ray is entirely outside aregion between said reflecting surface and said reference line.
 15. Theoptical device as defined in claim 12 wherein said irradiance is afunction additionally of brightness and said output angular range forthe angles between the normal to said reference line and right and leftedge rays arriving from said radiant source, said function of irradianceusable to iteratively tailor said reflector contour.
 16. The opticaldevice as defined in claim 15 wherein said collector of light is definedby tangent lines and a beginning known surface contour of said reflectorcontour generates a trailing edge ray which can be used in conjunctionwith said irradiance function I to calculate a leading edge ray.
 17. Theoptical device as defined in claim 12 wherein said reflecting surfacecomprises at least two reflector contours having a cusp therebetween.18. The optical device as defined in claim 17 wherein said two reflectorcontours are a minor image surface about said cusp.
 19. The opticaldevice as defined in claim 12 wherein dR(t)/dt is perpendicular toa(R(t))-u(t).
 20. The optical device as defined in claim 12 wherein saidreflector contour is comprised of a compound elliptical concentratorreflector.
 21. The optical device as defined in claim 12 wherein saidreflector contour is comprised of a compound hyperbolic concentratorreflector.
 22. The optical device as defined in claim 12 wherein theactual edge ray and the desired edge ray comprise an edge ray which hasundergone at most a single reflection and an edge ray which hasundergone multiple reflections, respectively.
 23. The optical device asdefined in claim 12 further including light refracting-medium means andwherein the desired edge ray is tailored into the actual edge ray bysaid light reflecting surface in combination with said lightrefracting-medium means.
 24. A method for producing a selected far fieldilluminance I output over an output angular range, comprising:providinglight having a characteristic irradiance, E; providing a lightreflecting surface having a reflecting contour and a family of edge raysdefined along a reference line; calculating said reflector contour withrespect to said reference line by using an expression of R(t) where t isa scalar parameter position:

    R(t)=k(t)+Du(t)

where k(t)=a parametrization of said reference line; D=a distance from apoint on said reference line to said reflecting surface along the adesired edge ray through the point; u=unit vector along the desireddirection of said desired edge ray; and said reflector contourreflecting said desired edge ray into an actual edge ray a(R(t)), wherea is a multidimensional unit vector pointing towards an edge of saidsource of light, and calculating said D by solving the expression:

    dD/dt=[(dk/dt)·(a-u)+D(du/dt)·a]/[(1-a)·u]

    dt 1-a·u.


25. The method as defined in claim 24 wherein said step of calculatingthe reflector contour comprises determining a value for the reflectorcontour at a first point relative to an observer of the light and thenutilizing the variation of irradiance E relative to the observer tocalculate a new point on the reflector contour by calculating a trailingedge ray of the light reflected from the first point, then using theirradiance E to calculate the direction of a leading edge ray and thenfurther using the expression dD/dt to determine a next portion of thereflector contour.
 26. The method as defined in claim 22 whereinirradiance E is,

    E=B(sin r.sub.R -sin r.sub.L)

where, B=brightness of the source and r_(R) and r_(L) are the anglesbetween the normal to the line R(t) and the respective right and leftedge rays striking the observer.
 27. The method as defined in claim 24wherein said reflector contour comprises a compound ellipticalconcentrator reflector and a compound hyperbolic concentrator reflector.28. The optical device as defined in claim 24 wherein the actual edgeray and the desired edge ray comprise an edge ray which has undergone atmost a single reflection and an edge ray which has undergone multiplereflections, respectively.
 29. The optical device as defined in claim 24further including light refracting-medium means and wherein the desirededge ray is tailored into the actual edge ray by said light reflectingsurface in combination with said light refracting-medium means.
 30. Amethod of collecting light of irradiance I from a radiant source presentover an output angular range, comprising:providing light having acharacteristic irradiance, E; providing a transducer collecting devicefor the light; providing a light reflecting surface having a reflectorcontour and a family of edge rays defined along a reference line, saidreflector contour defined in terms of said reference line by anexpression R(t) where t is a scalar parameter position, and:

    R(t)=k(t)+Du(t)

where k(t)=a parametrization of said reference line, D=a distance from apoint on said reference line to said reflecting surface along desirededge ray through the point; u=a unit vector along the desired directionof said desired edge ray; and said reflector contour reflecting saiddesired edge ray into actual edge ray, a(R(t)), where a is amultidimensional unit vector pointing towards an edge of said transducerof light and said D being the solutions of: ##EQU17##
 31. The method asdefined in claim 30 wherein said reflector contour is comprised of atleast one of a compound elliptical concentrator reflector.
 32. Theoptical device as defined in claim 30 wherein the actual edge ray andthe desired edge ray comprise an edge ray which has undergone at most asingle reflection and an edge ray which has undergone multiplereflections, respectively.
 33. The optical device as defined in claim 30further including light refracting-medium means and wherein the desirededge ray is tailored into the actual edge ray by said light reflectingsurface in combination with said light refracting-medium means.
 34. Anonimaging illumination optical device for producing a selected farfield illuminance I output over an angular range, comprising:a source oflight having a characteristic irradiance, E; light means for generatinga family of edge rays along a reference line, said light means includingat least one of (a) light refracting medium means and (b) a lightreflecting surface having a reflective contour and family of edge raysdefined in terms of said reference line by an expression R(t) where t isa scalar parameter position:

    a(t)=k(t)+Du(t)

where k(t)=a parametrization of said reference line; D=a distance from apoint on said reference line to said light means along a desired edgeray through the point; a unit vector along the desired direction of saiddesired edge ray; and said light means directing said desired edge rayinto an actual edge ray a(R(t)), where a is a multidimensional unitvector pointing towards an edge of said source of light.
 35. The opticaldevice as defined in claim 34 further including a reflecting surfacehaving a reflector contour and wherein the desired edge ray is tailoredinto the actual edge ray by said refraction means in combination withsaid reflector contour.
 36. The optical device as defined in claim 34wherein the actual edge ray and the desired edge ray comprise an edgeray which has undergone at most a single reflection and an edge raywhich has undergone multiple reflections, respectively.
 37. A nonimagingoptical device for collecting light or irradiance I from a radiantsource present over an output angular range, comprising:a source oflight having a characteristic irradiance, E; transducer means forcollecting the light from said source of light; light means foroperating on the light from said source of light to collect the lightassociated with a family of edge rays, said light means including atleast one of (a) light refracting medium means and (b) a lightreflecting surface having a reflector contour and the family of edgerays defined along a reference line, a point located on said referenceline and a desired edge ray passing through the point and extending tosaid reflecting surface and wherein t is a scaler parameter position andusing an expression R(t) to define said reflector contour:

    R(t)=k(t)+Du(t)

where k(t)=a parametrization of said reference line; D=a distance fromthe point on said reference line to at least one of said refractingmedium means and said reflecting surface along said desired edge raythrough the point; u=a unit vector along the desired direction of saiddesired edge ray; and said reflector contour and said light refractingmedium means reflecting and refracting, respectively, said desired edgeray into an actual edge ray a R(t)), where a is a multidimensional unitvector pointing towards an edge of said transducer means and wherein thedesired edge ray is tailored into the actual edge ray by at least one ofsaid reflector contour and said refracting medium means.
 38. The opticaldevice as defined in claim 37 wherein the actual edge ray and thedesired edge ray comprise an edge ray which has undergone at most asingle reflection and an edge ray which has undergone multiplereflections, respectively.